Solve Trigonometry Puzzle with P, S and H

In summary: That was a much more difficult problem than it seemed at first. In summary, the person is saying that it is not wholly trivial to derive the expressions for the other two sides in terms of P,S and H. They are forgetting their algebra, but the process is to solve an equation for x that takes into account the length of the side opposite of the given side. Finally, they give a simplified equation for x that can be solved by taking into account the length of the side opposite of the given side.
  • #1
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
10,123
137
Trigonometry "puzzle"

If we know the perimeter P of a triangle, a side S of it, and the height H down on the line going through S, then we have uniquely determined the triangle.

It is not wholly trivial to derive the expressions for the other two sides in terms of P,S and H.
 
Physics news on Phys.org
  • #2
I'm forgetting my algebra, but:

1) Essentially, we are given an ellipse, with the focii separated by S, a major axis of P, and if expressed as y=f(x), the corresponding y value of H for which we must solve.

2) The ellipse as a function (not purely as a function of x) might be:
sqrt((x+S/2)^2 + y^2) + sqrt((x-S/2)^2 + y^2) = P - S

3) Solve for x, given that y=H:
sqrt((x+S/2)^2 + H^2) + sqrt((x-S/2)^2 + H^2) = P - S
(I forget exactly how to go about this...)

4) With x in hand, it becomes trivial. The other sides would be:
Side 1 = sqrt((x+S/2)^2+y^2)
Side 2 = sqrt((x-S/2)^2+y^2)

DaveE
 
Last edited:
  • #3
The neat touch is, of course, to go over to a coordinate representation of the end points and utilize the properties of an ellipse.

The "brute force" approach to directly relate the SIDE LENGTHS to each other, say with a couple of Pythagorases will lead to ugly non-linear equations.

EDIT
NOTE:
Your initial formula is wrong. On your right-hand side, it should say P-S, not P!
(The sum of the two other sides is constant)
 
Last edited:
  • #4
arildno said:
The neat touch is, of course, to go over to a coordinate representation of the end points and utilize the properties of an ellipse.

Heh, that's all well and good if you remember your ellipse properties, but for those of us who're 14 years out of their geometry classes, we're stuck with brute force :) (short of going and looking up all those old properties)

arildno said:
The "brute force" approach to directly relate the SIDE LENGTHS to each other, say with a couple of Pythagorases will lead to ugly non-linear equations.

Out of curiosity, how *would* you go about solving that ugly equation for x? Anyone?

arildno said:
Your initial formula is wrong. On your right-hand side, it should say P-S, not P!

Ooops, fixed!

DaveE
 
  • #5
Hmm..you gave the simple ellipse approach in your initial post.
Now, as for solving for x, the simplest way is like this:
[tex]\sqrt{(x-\frac{S}{2})^{2}+H^{2}}=(P-S)-\sqrt{(x+\frac{S}{2})^{2}+H^{2}}[/tex]
Square this expression, gaining:
[tex](x-\frac{S}{2})^{2}+H^{2}=(x+\frac{S}{2})^{2}+(P-S)^{2}-2(P-S)\sqrt{(x+\frac{S}{2})^{2}+H^{2}}[/tex]
Simplify to:
[tex]\sqrt{(x+\frac{S}{2})^{2}+H^{2}}=\frac{xS}{(P-S)}+\frac{(P-S)}{2}[/tex]
Re-square:
[tex](x+\frac{S}{2})^{2}+H^{2}=\frac{S^{2}}{(P-S)^{2}}x^{2}+xs+\frac{(P-S)^{2}}{4}[/tex]
Simplify this to:
[tex](1-\frac{S^{2}}{(P-S)^{2}})x^{2}=\frac{P(P-2S)}{4}-H^{2}[/tex]
From which we gain the positive solution:
[tex]x=\frac{(P-S)}{2}\sqrt{1-\epsilon},\epsilon=\frac{4H^{2}}{P(P-2S)}[/tex]
 
  • #6
Cool question man.
 

What is the purpose of solving trigonometry puzzles with P, S, and H?

The purpose of solving trigonometry puzzles with P, S, and H is to practice and apply trigonometric principles, such as calculating angles and side lengths in right triangles. It also helps to improve problem-solving skills and logical thinking.

What do P, S, and H represent in trigonometry puzzles?

P, S, and H typically represent the three sides of a right triangle - the perpendicular side (P), the base side (B), and the hypotenuse (H). In some puzzles, they may also represent angles or other measurements.

What are some common strategies for solving trigonometry puzzles with P, S, and H?

Some common strategies for solving trigonometry puzzles with P, S, and H include using the Pythagorean theorem, trigonometric ratios (sine, cosine, tangent), and special right triangle properties. It is also helpful to draw diagrams and use logical reasoning to approach the problem.

Are there any real-world applications for solving trigonometry puzzles with P, S, and H?

Yes, trigonometry puzzles with P, S, and H have many real-world applications, such as in architecture, engineering, navigation, and physics. They are also commonly used in solving problems related to triangles and angles in various fields.

Can trigonometry puzzles with P, S, and H be solved using a calculator?

Yes, trigonometry puzzles with P, S, and H can be solved using a calculator, especially for more complex problems. However, it is still important to understand the concepts and principles behind the calculations in order to fully grasp the solution.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
  • General Math
Replies
1
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
746
  • Precalculus Mathematics Homework Help
Replies
2
Views
900
  • Advanced Physics Homework Help
Replies
5
Views
956
  • Thermodynamics
Replies
4
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
8
Views
751
Replies
24
Views
2K
  • Introductory Physics Homework Help
Replies
14
Views
3K
  • Introductory Physics Homework Help
Replies
13
Views
1K
Back
Top